#### Transcript Slide 1

```Chapter 11
Vibrations and Waves
11-1 Simple Harmonic Motion
If an object vibrates or
oscillates back and forth
over the same path,
each cycle taking the
same amount of time,
the motion is called
periodic. The mass and
spring system is a
useful model for a
periodic system.
11-1 Simple Harmonic Motion
We assume that the surface is frictionless.
There is a point where the spring is neither
stretched nor compressed; this is the
equilibrium position. We measure
displacement from that point (x = 0 on the
previous figure).
The force exerted by the spring depends on
the displacement:
(11-1)
11-1 Simple Harmonic Motion
• The minus sign on the force indicates that it
is a restoring force – it is directed to restore
the mass to its equilibrium position.
• k is the spring constant
• The force is not constant, so the acceleration
is not constant either
11-1 Simple Harmonic Motion
• Displacement is measured from
the equilibrium point
• Amplitude is the maximum
displacement
• A cycle is a full to-and-fro
motion; this figure shows half a
cycle
• Period is the time required to
complete one cycle
• Frequency is the number of
cycles completed per second
11-1 Simple Harmonic Motion
If the spring is hung
vertically, the only change
is in the equilibrium
position, which is at the
point where the spring
force equals the
gravitational force.
11-1 Simple Harmonic Motion
Any vibrating system where the restoring
force is proportional to the negative of
the displacement is in simple harmonic
motion (SHM), and is often called a
simple harmonic oscillator.
11-2 Energy in the Simple Harmonic Oscillator
We already know that the potential energy of a
spring is given by:
The total mechanical energy is then:
(11-3)
The total mechanical energy will be
conserved, as we are assuming the system
is frictionless.
11-2 Energy in the Simple
Harmonic Oscillator
If the mass is at the limits of
its motion, the energy is all
potential.
If the mass is at the
equilibrium point, the
energy is all kinetic.
We know what the
potential energy is at the
turning points =
Ei = Ef
11-3 The Period and
Sinusoidal Nature of SHM
SHM can be modeled as the:
the projection onto the x axis
of an object moving in a circle
of radius A at a constant speed
This is identical to SHM.
The Period and Sinusoidal Nature of SHM
2A
vmax 
T
2A
T 
vmax
1 / 2kA2  1 / 2m v2
A
m
 
v
k
m
 T  2
k
Therefore, we can use
the period and
frequency of a particle
moving in a circle to
find the period and
frequency:
Example: A SHM oscillator has a mass of 1.5 kg and a
spring constant of 5 N/m. The initial amplitude of the
system is 0.5 m.
A.Calculate the maximum velocity of the system.
B.Calculate the velocity at x = 0.2 m.
C.Calculate the period of the oscillator.
D.Calculate the frequency of the oscillator
A.
1
1
1
2
2
2
m v  2 kx  2 kA
2
1
2
m v  kA2
2
1
2
kA2
5N/m(0.5m) 2
v

 0.91m/s
m
1.5kg
B.Calculate the velocity at x = 0.2 m.
1
2
1
2
m v2  12 kx2  12 kA2
m v2   12 kA2  12 kx2
k ( A2  x 2 )
v
m
k ( A2  x 2 )
5N/m[(0.5m) 2  (0.2m) 2 ]
v

 0.84m/s
m
1.5kg
C.Calculate the period of the oscillator.
m
1.5kg
T  2
 2
 3.44s
k
5N/m
D.Calculate the frequency of the oscillator
1
1
f  
 0.29Hz
T 3.44s
11-4 The Simple Pendulum
In order to be in SHM, the
restoring force must be
proportional to the negative of
the displacement. Here we
have:
which is proportional to sin θ
and not to θ itself.
However, if the
angle is small,
sin θ ≈ θ.
Simple Pendulum
•For SHM, the restoring force:
F = -kx
•For a pendulum, the restoring force:
F = -mgsinθ
•For small angles:
F = -mgθ
But θ = x/L  F = -mgx/L
 k = mg/L
T = 2
m
k
= 2
m
= 2 L
mg/L
g
11-4 The Simple Pendulum
The period and frequency are:
11-3 The Period and Sinusoidal Nature of SHM
The top curve is a
graph of the previous
equation.
The bottom curve is
the same, but shifted
¼ period so that it is
a sine function rather
than a cosine.
11-3 The Period and
Sinusoidal Nature of
SHM
The velocity and acceleration can
be calculated as functions of
time; the results are below, and
are plotted at left.
11-3 The Period and Sinusoidal Nature of SHM
We can similarly find the position as a function of time:
v = -Asin(2ft)
a = -A2cos(2ft)
 = 2f
A 0.9 kg mass vibrates according to the equation:
x = 1.2cos3t where x is in meters and t is in seconds.
Determine: a) amplitude b) frequency c) total energy
d) the kinetic and potential energies at x = 0.3m e) vmax f) amax
a. A  1.2m
b.   2f

3
f 

 0.477s 1
2 2
c. Etotal  12 kA2
Find k
m
T  2
k
1
1 k
1 k
f  

T 2 m 2 m
k  (2f ) 2 m  (2 (0.477s 1 ))2 (0.9kg )  1.29N/m
 Etotal  12 (1.29N/m)1.2m 2  0.929J
d. PE0.3m  12 kx 2  12 (1.29N/m)(0.3m)2  0.058J
KE0.3m  Etotal  PE0.3m  0.929J  0.058J  0.871J
f. vmax  A  (1.2m)(3rad/s)  3.6m/s
a max  A 2  (1.2m)(3rad/s) 2  10.8m/s2
11-5 Damped Harmonic Motion
Damped harmonic motion is harmonic
motion with a frictional or drag force. If the
damping is small, we can treat it as an
“envelope” that modifies the undamped
oscillation.
11-5 Damped Harmonic Motion
However, if the damping is
large, it no longer resembles
SHM at all.
A: underdamping: there are
a few small oscillations
before the oscillator comes
to rest.
B: critical damping: this is the fastest way to get to
equilibrium.
C: overdamping: the system is slowed so much
that it takes a long time to get to equilibrium.
11-5 Damped Harmonic Motion
There are systems where damping is unwanted,
such as clocks and watches.
Then there are systems in which it is wanted, and
often needs to be as close to critical damping as
possible, such as automobile shock absorbers
and earthquake protection for buildings.
11-7 Wave Motion
A wave travels
along its medium,
but the individual
particles just move
up and down.
11-7 Wave Motion
All types of traveling waves transport energy.
Study of a single wave
pulse shows that it is
begun with a vibration
and transmitted through
internal forces in the
medium.
Continuous waves start
with vibrations too. If the
vibration is SHM, then the
wave will be sinusoidal.
11-7 Wave Motion
Wave characteristics:
• Amplitude, A
• Wavelength, λ
• Frequency f and period T
• Wave velocity
(11-12)
11-8 Types of Waves: Transverse and
Longitudinal
The motion of particles in a wave can either be
perpendicular to the wave direction (transverse) or
parallel to it (longitudinal).
11-8 Types of Waves: Transverse and
Longitudinal
Sound waves are longitudinal waves:
11-9 Energy Transported by Waves
If a wave is able to spread out threedimensionally from its source, and the medium is
uniform, the wave is spherical.
Just from geometrical
considerations, as long as
the power output is
constant, we see:
(11-16b)
11-11 Reflection and Transmission of Waves
A wave reaching the end
of its medium, but where
the medium is still free
to move, will be reflected
(b), and its reflection will
be upright.
A wave hitting an obstacle will be
reflected (a), and its reflection will be
inverted.
11-11 Reflection and Transmission of Waves
A wave encountering a denser medium will be partly
reflected and partly transmitted; if the wave speed is
less in the denser medium, the wavelength will be
shorter.
11-12 Interference; Principle of Superposition
These figures show the sum of two waves. In (a)
destructively; and in (c) they add partially
destructively.
11-13 Standing Waves; Resonance
Standing waves occur
when both ends of a
string are fixed. In that
case, only waves which
are motionless at the
ends of the string can
persist. There are nodes,
where the amplitude is
always zero, and
antinodes, where the
amplitude varies from
zero to the maximum
value.
11-13 Standing Waves; Resonance
The frequencies of the
standing waves on a
particular string are called
resonant frequencies.
They are also referred to as
the fundamental and
harmonics.
11-11 Reflection and Transmission of Waves
Two- or three-dimensional waves can be
represented by wave fronts, which are curves
of surfaces where all the waves have the same
phase.
Lines perpendicular to
the wave fronts are
called rays; they point in
the direction of
propagation of the wave.
11-11 Reflection and Transmission of Waves
The law of reflection: the angle of incidence
equals the angle of reflection.
11-12 Interference; Principle of Superposition
The superposition principle says that when two waves
pass through the same point, the displacement is the
arithmetic sum of the individual displacements.
In the figure below, (a) exhibits destructive interference
and (b) exhibits constructive interference.
11-13 Standing Waves; Resonance
The wavelengths and frequencies of standing
waves are:
(11-19a)
(11-19b)
11-14 Refraction
If the wave enters a medium where the wave
speed is different, it will be refracted – its wave
fronts and rays will change direction.
We can calculate the angle of
refraction, which depends on
both wave speeds:
(11-20)
11-14 Refraction
The law of refraction works both ways – a wave
going from a slower medium to a faster one
would follow the red line in the other direction.
11-15 Diffraction
When waves encounter
an obstacle, they bend
around it, leaving a
called diffraction.
11-15 Diffraction
The amount of diffraction depends on the size of
the obstacle compared to the wavelength. If the
obstacle is much smaller than the wavelength,
the wave is barely affected (a). If the object is
comparable to, or larger than, the wavelength,
diffraction is much more significant (b, c, d).
11-10 Intensity Related to Amplitude and
Frequency
By looking at the
energy of a particle of
matter in the medium
of the wave, we find:
Then, assuming the entire medium has the same
density, we find:
(11-17)
Therefore, the intensity is proportional to the
square of the frequency and to the square of the
amplitude.
11-16 Mathematical Representation of a
Traveling Wave
To the left, we have a
snapshot of a traveling
wave at a single point
in time. Below left, the
same wave is shown
traveling.
11-16 Mathematical Representation of a
Traveling Wave
A full mathematical description of the wave
describes the displacement of any point as a
function of both distance and time:
(11-22)
Summary of Chapter 11
• For SHM, the restoring force is proportional to
the displacement.
• The period is the time required for one cycle,
and the frequency is the number of cycles per
second.
• Period for a mass on a spring:
• SHM is sinusoidal.
• During SHM, the total energy is continually
changing from kinetic to potential and back.
Summary of Chapter 11
• A simple pendulum approximates SHM if its
amplitude is not large. Its period in that case is:
• When friction is present, the motion is
damped.
• If an oscillating force is applied to a SHO, its
amplitude depends on how close to the natural
frequency the driving frequency is. If it is
close, the amplitude becomes quite large. This
is called resonance.
Summary of Chapter 11
• Vibrating objects are sources of waves, which
may be either a pulse or continuous.
• Wavelength: distance between successive
crests.
• Frequency: number of crests that pass a given
point per unit time.
• Amplitude: maximum height of crest.
• Wave velocity:
Summary of Chapter 11
• Vibrating objects are sources of waves, which
may be either a pulse or continuous.
• Wavelength: distance between successive
crests
• Frequency: number of crests that pass a given
point per unit time
• Amplitude: maximum height of crest
• Wave velocity:
Summary of Chapter 11
• Transverse wave: oscillations perpendicular to
direction of wave motion.
• Longitudinal wave: oscillations parallel to
direction of wave motion.
• Intensity: energy per unit time crossing unit
area (W/m2):
• Angle of reflection is equal to angle of
incidence.
Summary of Chapter 11
• When two waves pass through the same region
of space, they interfere. Interference may be
either constructive or destructive.
• Standing waves can be produced on a string
with both ends fixed. The waves that persist are
at the resonant frequencies.
• Nodes occur where there is no motion;
antinodes where the amplitude is maximum.
• Waves refract when entering a medium of
different wave speed, and diffract around
obstacles.
11-3 The Period and
Sinusoidal Nature of
SHM
The velocity and acceleration can
be calculated as functions of
time; the results are below, and
are plotted at left.
(11-9)
(11-10)
11-6 Forced Vibrations; Resonance
Forced vibrations occur when there is a
periodic driving force. This force may or may
not have the same period as the natural
frequency of the system.
If the frequency is the same as the natural
frequency, the amplitude becomes quite large.
This is called resonance.
11-6 Forced Vibrations; Resonance
The sharpness of the
resonant peak depends
on the damping. If the
damping is small (A), it
can be quite sharp; if
the damping is larger
(B), it is less sharp.
Like damping, resonance can be wanted or
11-9 Energy Transported by Waves
Just as with the oscillation that starts it, the
energy transported by a wave is proportional to
the square of the amplitude.
Definition of intensity:
The intensity is also proportional to the
square of the amplitude:
(11-15)
11-8 Types of Waves: Transverse and
Longitudinal
Earthquakes produce both longitudinal and
transverse waves. Both types can travel through
solid material, but only longitudinal waves can
propagate through a fluid – in the transverse
direction, a fluid has no restoring force.
Surface waves are waves that travel along the
boundary between two media.
Units of Chapter 11
•Simple Harmonic Motion
•Energy in the Simple Harmonic Oscillator
•The Period and Sinusoidal Nature of SHM
•The Simple Pendulum
•Damped Harmonic Motion
•Forced Vibrations; Resonance
•Wave Motion
•Types of Waves: Transverse and Longitudinal
Units of Chapter 11
•Energy Transported by Waves
•Intensity Related to Amplitude and Frequency
•Reflection and Transmission of Waves
•Interference; Principle of Superposition
•Standing Waves; Resonance
•Refraction
•Diffraction
•Mathematical Representation of a Traveling Wave
```